For an elliptic curve \(E\)  with algebraic rank zero, denote the Euler characteristic of the Selmer group of \(E\)  over the cyclotomic \(\mathbb {Z}_p\) -extension by \(\chi (\Gamma , E)\)  (cf. [1]}). When \(E\)  has algebraic rank zero, the Euler-characteristic \(\chi (\Gamma , E)\)  coincides with the truncated Euler-characteristic \(\chi _t(\Gamma ,E)\) . Shekhar and the second author [2]} deduced congruence results for the Euler characteristic of pairs of congruent elliptic curves \(E_1\)  and \(E_2\)  of rank zero. In particular, they deduce that if \(\chi (\Gamma , E_1)=1\) , then \(\chi (\Gamma , E_2)=1\) . Let \(m\)  be a \(p\) -power free integer coprime to \(p\)  as well as to the conductors of both elliptic curves. For the false Tate curve extension defined by \(\mathbb {Q}_{\infty }=\mathbb {Q}(\mu _{p^{\infty }}, m^{\frac{1}{p^{\infty }}})\)  it is shown in [2]} if \(\chi (G, E_1)=1\)  then so is \(\chi (G,E_2)=1\) . In this article, analogous results for truncated Euler characteristics in the positive rank setting are proved. We stress that our methods are different from those in [2]}, and hence yield another proof of the results in the rank zero case as well. Generalizations of the results in this paper to modular forms and abelian varieties over arbitrary number fields are currently being investigated.
